Consider the tests
\( F_\varepsilon (e^{it}) := \dfrac{1 - \varepsilon e^{it}}{(1 - \varepsilon e^{-it})^{1 - \frac{2}{p^{\prime }}}}, \quad \varepsilon > 0.\) 
It is natural to consider this family of functions here because it shows that, if true, the exponent \(\frac{4}{p}\)  in Conjecture REF  cannot be improved [1]}. It is a straightforward computation to check that \(\Vert  P_+ F_\varepsilon \Vert _{A^2_\alpha } \le \Vert  F_\varepsilon \Vert _{L^{p^{\prime }}}\)  for all \(\varepsilon > 0\)  implies that \(\alpha \ge \frac{p}{2} - 2\) , and hence we have proved the necessity of this condition.
Take \(F \in L^{p^{\prime }} (\) . Since the dual of \(A^2_{\frac{p}{2} - 2}\)  with respect to the \(H^2\) -pairing is \(D_{\frac{p}{2}}\)  and \(P_+\)  is self-adjoint, we have that
\(\left| \left\langle P_+ F, g\right\rangle _{H^2} \right| = \left| \left\langle F, g \right\rangle _{L^2 (} \right| \le \Vert  F \Vert _{L^{p^{\prime }}} \Vert  g \Vert _{H^p} \le \Vert  F \Vert _{L^{p^{\prime }}} \Vert  g \Vert _{D_{\frac{p}{2}}},\) 
for all \(g \in D_{\frac{p}{2}}\)  and hence \(P_+\)  is contractive from \(L^{p^{\prime }}(\)  to \(A^2_{\frac{p}{2}-2}\) . Thus, Corollary REF  yields that
\(\Vert  P_+ F \Vert _{A^2_\alpha } \le \Vert  P_+ F \Vert _{A^2_{\frac{p}{2} - 2}} \le \Vert  F \Vert _{L^{p^{\prime }}}, \quad \forall F \in L^{p^{\prime }} (,\) 
if \(\alpha \ge \frac{p}{2} - 2\)  and then we have completed the proof.
